Maximum principle for stochastic optimal control problem under convex expectation

In this paper, we study a stochastic optimal control problem under a type of consistent convex expectation dominated by G-expectation. By the separation theorem for convex sets, we get the representation theorems for this convex expectation and conditional convex expectation. Based on these results, we obtain the variational equation for cost functional by weak convergence and discretization methods. Furthermore, we establish the maximum principle which is sufficient under usual convex assumptions. Finally, we study the linear quadratic control problem by using the obtained maximum principle.